Final answer:
To determine the expected sum of rolls from two weighted dice with the number 5 twice as likely, calculate the probability of each roll and its product with the roll value, then sum these expected values.
Step-by-step explanation:
The subject of this question is mathematics, specifically focused on the concept of probability with weighted dice. When considering dice where the number 5 appears twice as often as all other numbers, we have a non-standard probability distribution for the dice rolls. To find the expected sum of two such weighted dice, we firstly need to determine the probability of each roll. Since there are six faces on a die and the number 5 occurs twice as often, the probability of rolling a 5 is 2 / 7, while the probability of rolling any other number is 1 / 7. The sum for each possible roll must be multiplied by its probability and then all those expected values must be added together to find the overall expected sum.
To illustrate, the expected value (EV) of rolling a 5 on a weighted die is (5 * (2/7)) and for a 3, it would be (3 * (1/7)). For two dice, you'd find the sum of the products of the sums of each pair of numbers and their associated probabilities (e.g., for a roll of 5 and 3, the expected value would be (5+3) * (2/7 * 1/7)). Finally, you add together the EVs for all possible sums.